The Stein kernel, which shall be defined later, has been shown [1]}, [2]} to provide another way to construct analytic bounds on the Wasserstein metric between two known distributions. For this reason, we shall be utilising the Stein kernel to bound the Wasserstein metric. This avoids the need to bound the solution to the Stein equation directly, which can sometimes yield large bounds. For example, when looking at the von-Mises distribution \(X \sim VM( 0,\kappa )\) , one will find that the solution to the Stein equation has the form (up to a constant proportional to \(e^{-\kappa \cos x}\) ),
\( f_h(x) = e^{-\kappa \cos (x)} \int _{-\pi }^x (h(u)-\mathbb {E}[h(X)])e^{\kappa \cos (u)} du.\) 
